We investigate in this chapter the general problem of the interaction of an atom (or a molecule) and a classical electromagnetic field. Its importance derives firstly from the fact that a large part of our knowledge of atoms is obtained from a study of the radiation they absorb or emit (we shall consider equally visible radiation and radio-frequency and X-rays) and, secondly, that the interaction with matter modifies the propagation of the electromagnetic field itself, notably through absorption, refraction or scattering. The interactions of atoms with light therefore encompass a vast range of physical effects that we could not hope to cover adequately in a single chapter. In this chapter we shall therefore present the fundamental features of the interaction of an atom, which will be treated quantum mechanically, with a classical electromagnetic field, that is an electromagnetic field described by real electric and magnetic vectors obeying Maxwell's equations.

A rigorous description of the atom–light interaction would have to take into account the quantum nature of light. This we shall leave to Chapter 6. It turns out, nevertheless, that many important results can be obtained from the semi-classical viewpoint adopted here (although it is perhaps more appropriate to refer to this treatment as ‘semi-quantized’).

Since its invention in 1960, the laser has revolutionized both the study of optics and our understanding of the nature of light, prompting the emergence of a new field, quantum optics. Actually, it took decades until the words quantum optics took their current precise meaning, referring to phenomena which can be understood only by quantizing the electromagnetic field describing light. Surprisingly enough, such quantum optics phenomena hardly existed at the time that the laser was invented, and almost all optics effects could be fully understood by describing light as a classical electromagnetic field; the laser was no exception. As a matter of fact, to understand how a laser works, it suffices to use the semi-classical description of matter–light interaction, where the laser amplifying medium, made of atoms, molecules, ions or semi-conductors, is given a quantum mechanical treatment, but light itself is described by classical electromagnetic waves.

The first part of our book is devoted to presentation of the semi-classical approach and its use in describing various optical phenomena. It includes an elementary exposition of the physics of lasers, and some applications of this ubiquitous device. After recalling in Chapter 1 some basic results of the quantum mechanical description of interaction induced transitions between the atomic energy levels, we use these results in Chapter 2 to show how the interaction of a quantized atom with a classical electromagnetic wave leads to absorption or stimulated emission, and to derive the process of laser amplification that happens when a wave propagates in an inverted medium.

The arguments of Chapter 2, as well as of those of subsequent chapters, have as their foundation the formalism based on the state vector of a system of which the evolution is described by the Schrödinger equation. In fact, such an approach is badly suited to the case in which the coupling between an atom and its environment (for example through collisions with other atoms or spontaneous emission into formerly empty modes of the electromagnetic field) cannot be neglected. If the correlations induced by these interactions between the atom and its environment do not concern us and we are only interested in the evolution of the atom, the formalism of the density matrix must be employed. This provides a description at all times of the state of the atom, although a state vector for the atom alone cannot be defined. In this formalism the effect of the environment on the atom is accounted for by the introduction of suitable relaxation terms (Section 2C.1) in the equation of evolution of the density matrix. An important application of the density matrix is to the case of a two-level atomic system for which the relaxation terms lead to its deexcitation to a level of lower energy. We shall show that in this case the density matrix can be represented by a vector, known as the Bloch vector, which will allow us to give simple geometrical pictures of the evolution of the system.

In this work we shall study the interaction of matter and light. In so doing we shall rely heavily on the description of such processes provided by quantum mechanics. This appears on a number of levels: firstly, a quantum description of matter is indispensable if one wants to understand on the microscopic scale the different kinds of interaction processes that can occur. Secondly, a quantum description of light often turns out to be useful, sometimes necessary, to better understand these processes. We shall study phenomena such as spontaneous emission, which can only be properly treated by a theory taking into account the quantum nature of both light and matter.

In the following chapters we shall address, amongst others, the following question: ‘given an atom prepared at a given time in a particular state and subjected from this time onwards to electromagnetic radiation, what is the state of the atom and radiation at any later moment in time?’ In order to be able to answer this question it will be necessary for us to know how to calculate the evolution of a quantum system in a small number of typical situations. These methods we shall demonstrate in the first chapter.

The evolution of the coupled atom–light system depends on the temporal dependence of the applied light field, which could, for example, be applied from a given moment and thereafter remain unchanged in intensity, or, perhaps, be appreciable only for a finite period of time (pulsed excitation).

In Section 8.3.9, we mentioned that the recoil temperature was not the ultimate barrier to cooling. Several mechanisms have been conceived and implemented to concentrate the atomic velocity distribution in a range of values of width less than VR. In contrast to Doppler or Sisyphus cooling, these mechanisms do not appeal to a friction force, but rather to an optical pumping process (see Complement 2B), able to accumulate the atoms in this narrow range of velocities. We now present one such mechanism called ‘velocity-selective coherent population trapping’, considering only the case where the motion occurs in one space direction (the Oz-axis). Note, however, that the method generalizes to two or three dimensions.

Coherent population trapping

Coherent population trapping is a phenomenon that occurs when an atom has a system of three energy levels in the ∧ configuration shown in Figure 8A.1, where each leg of the ∧ interacts with a laser L− or L+. This was discussed in detail in Section 2D.2 of Complement 2D. Here we consider the case where the two ground states g+ and L−, have exactly the same energy, and the two lasers have frequencies ω+ and ω−, very close to the resonance frequency ω0. In this situation, the selectivity of the interactions is obtained via the polarization selection rules (see Complement 2B): if g−, g+ and e have magnetic quantum numbers m = −1, +1 and 0, respectively, the laser wave L− is right-circularly polarized (σ+ polarization), while the laser wave L+ is left-circularly polarized (σ− polarization).